Index is multiplicative for profinite groups

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Statement

Suppose G is a profinite group and K,H are closed subgroups of G with K \le H. Note that K automatically becomes closed in H. (Conversely, K being closed in H and H being closed in G would imply that K is closed in G). Then, we have:

[G:K] = [G:H][H:K]

where [G:K] denote the respective values for the index of a closed subgroup in a profinite group where the subgroup is K and the group is G. Similarly for [G:H] (index of subgroup H in group G) and [H:K] (index of subgroup K in group H).

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